In mathematics, the cyclic category or cycle category or category of cycles is a category of finite cyclically ordered sets and degree-1 maps between them. It was introduced by .
The cyclic category ÃÂ has one object ÃÂ<sub>n</sub> for each natural number n = 0, 1, 2, ...
The morphisms from ÃÂ<sub>m</sub> to ÃÂ<sub>n</sub> are represented by increasing functions f from the integers to the integers, such that f(x+m+1) = f(x)+n+1, where two functions f and g represent the same morphism when their difference is divisible by n+1.
Informally, the morphisms from ÃÂ<sub>m</sub> to ÃÂ<sub>n</sub> can be thought of as maps of (oriented) necklaces with m+1 and n+1 beads. More precisely, the morphisms can be identified with homotopy classes of degree 1 increasing maps from S<sup>1</sup> to itself that map the subgroup Z/(m+1)Z to Z/(n+1)Z.
The number of morphisms from ÃÂ<sub>m</sub> to ÃÂ<sub>n</sub> is (m+n+1)!/m!n!.
The cyclic category is self dual.
The classifying space BÃÂ of the cyclic category is a classifying space BS<sup>1</sup>of the circle group S<sup>1</sup>.
A cyclic set is a contravariant functor from the cyclic category to sets. More generally a cyclic object in a category C is a contravariant functor from the cyclic category to C.